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Mirrors > Home > MPE Home > Th. List > Mathboxes > ontgsucval | Structured version Visualization version GIF version |
Description: The topology generated from a successor ordinal number is itself. (Contributed by Chen-Pang He, 11-Oct-2015.) |
Ref | Expression |
---|---|
ontgsucval | ⊢ (𝐴 ∈ On → (topGen‘suc 𝐴) = suc 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | onsuc 7824 | . . 3 ⊢ (𝐴 ∈ On → suc 𝐴 ∈ On) | |
2 | ontgval 36374 | . . 3 ⊢ (suc 𝐴 ∈ On → (topGen‘suc 𝐴) = suc ∪ suc 𝐴) | |
3 | 1, 2 | syl 17 | . 2 ⊢ (𝐴 ∈ On → (topGen‘suc 𝐴) = suc ∪ suc 𝐴) |
4 | eloni 6390 | . . . 4 ⊢ (𝐴 ∈ On → Ord 𝐴) | |
5 | ordunisuc 7845 | . . . 4 ⊢ (Ord 𝐴 → ∪ suc 𝐴 = 𝐴) | |
6 | 4, 5 | syl 17 | . . 3 ⊢ (𝐴 ∈ On → ∪ suc 𝐴 = 𝐴) |
7 | suceq 6446 | . . 3 ⊢ (∪ suc 𝐴 = 𝐴 → suc ∪ suc 𝐴 = suc 𝐴) | |
8 | 6, 7 | syl 17 | . 2 ⊢ (𝐴 ∈ On → suc ∪ suc 𝐴 = suc 𝐴) |
9 | 3, 8 | eqtrd 2773 | 1 ⊢ (𝐴 ∈ On → (topGen‘suc 𝐴) = suc 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1535 ∈ wcel 2104 ∪ cuni 4914 Ord word 6379 Oncon0 6380 suc csuc 6382 ‘cfv 6558 topGenctg 17473 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1963 ax-7 2003 ax-8 2106 ax-9 2114 ax-10 2137 ax-11 2153 ax-12 2173 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pow 5366 ax-pr 5430 ax-un 7747 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1538 df-fal 1548 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2536 df-eu 2565 df-clab 2711 df-cleq 2725 df-clel 2812 df-nfc 2888 df-ne 2937 df-ral 3058 df-rex 3067 df-rab 3433 df-v 3479 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4915 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5635 df-we 5637 df-xp 5689 df-rel 5690 df-cnv 5691 df-co 5692 df-dm 5693 df-ord 6383 df-on 6384 df-suc 6386 df-iota 6510 df-fun 6560 df-fv 6566 df-topgen 17479 |
This theorem is referenced by: onsuctop 36376 |
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